Effective Hamiltonian of Liquid-Vapor Curved Interfaces in Mean Field

TL;DR

This paper derives an exact mean-field expression for the grand potential of a curved liquid-vapor interface, connecting microscopic density functional theory with the Helfrich model through curvature expansion.

Contribution

It introduces a simple, exact mean-field level expression for the grand potential of curved interfaces, linking microscopic density profiles to macroscopic curvature models.

Findings

Grand potential expressed in terms of density profile and interactions.

Expansion in curvatures aligns with Helfrich model.

Consistency established at second order in curvature.

Abstract

We analyze a one-component simple fluid in a liquid-vapor coexistence state, which forms an arbitrarily curved interface. By using an approach based on density functional theory, we obtain an exact and simple expression for the grand potential at the level of mean field approximation that depends on the density profile and the short-range interaction potential. By introducing the step-function approximation for the density profile, and using general geometric arguments, we expand the grand potential in powers of the principal curvatures of the surface and find consistency with the Helfrich phenomenological model in the second order approximation.